Why not take the train which stops right by the ship? It will take you to Liverpool St.Station in London,in one hour and fifteen minutes(some trains are direct, some have one change but it is a very short wait) It will cost around Ã‚Â£20 one way and is a great fast service- no need to pay more.....
We took the train from Liverpool Station in London to Harwich last fall, and have taken the cruise ship transfers in the past. The National Rail trip has a station adjacent to the cruise terminal, and you can, of course, get a taxi at Liverpool to your destination. You can get all the rail schedule and price information on their web site, and get tickets if you wish. I would use the train in the future myself.
Or try http://www.cruisedrive.com They were recommended to me by someone on the board.
They are a bit pricey though - at least for a family of 4 it was going to be well over $650 for a round-trip transfer from London to Southampton and back to LHR.